Semidefinite Programming Certificates for Synchronization of Kuramoto Oscillators on Arcs
It offers a novel certification method for synchronization in coupled oscillator systems, which is important for power grids and biological networks, but the results are incremental as they extend existing techniques to a specific class of models.
The paper proposes a semidefinite programming approach to certify local phase synchronization in a class of Kuramoto oscillator models with general coupling functions, for all initial conditions on an arc. The method provides certificates for stability of the phase-difference system, implying synchronization.
A class of Kuramoto models with a general coupling function that can be expressed in terms of a finite number of harmonics, each comprising sinusoidal terms, is studied. We propose a novel approach for certifying local phase synchronization in this class for all initial conditions lying on an arc. The trace parametrization property and Gram matrix representation of a trigonometric polynomial are utilized along with Putinar's Positivstellensatz to obtain semidefinite programming certificates for the stability of the phase-difference system, which in turn implies synchronization of the original system. The results can be extended to any system of coupled oscillators where the forward-invariance on arcs can be established.